Boundary behavior p-harmonic functions in the Heisenberg group

We study the boundary behavior of nonnegative p-harmonic functions which vanish on a portion of the boundary of a domain in the Heisenberg group H^n. Our main results are: 1) An estimate from above which shows that, under suitable geometric assumptions on the relevant domain, such a p-harmonic function vanishes at most linearly with respect to the sub-Riemannian distance to the boundary. 2) An estimate from below which shows that for a (Euclidean) C^{1,1} domain, away from the characteristic set, such a p-harmonic function vanishes exactly like the distance to the boundary. By combining 1) and 2) we obtain a comparison theorem stating that, at least away from the characteristic set, any two such p-harmonic functions must vanish at the same rate

Potential theory and harmonic analysis methods for quasilinear and Hessian equations

The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file.Title from title screen of research.pdf file viewed on (February 28, 2007)Vita.Thesis (Ph.D.) University of Missouri-Columbia 2006.The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:-[delta]pu = uq + [mu], Fk[-u] = uq + [mu], u [greater than or equal to] 0, on Rn, or on a bounded domain [omega] [subset of or implied by] Rn. Here [delta]p is the p-Laplacian defined by [delta]pu = div ([delta]u [delta]u p-2), and Fk[u] is the k-Hessian defined as the sum of k x k principal minors of the Hessian matrix D2u (k = 1, 2, . . . , n); [mu] is a nonnegative measurable function (or measure) on [omega]. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data [mu] [such that element of] Ls([omega]), s [greater than] 1. Such results are deduced from our existence criteria with the sharp exponents s = (n(q-p+1)) / pq for the first equation, and s = (n(q-k) q-k) / 2kq for the second one. Furthermore, a complete characterization of removable singularities for each corresponding homogeneous equation is given as a consequence of our solvability results.Includes bibliographical reference

Existence and regularity estimates for quasilinear equations with measure data: the case 1<p≤3n−22n−11<p\leq \frac{3n-2}{2n-1}

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-{\rm div} (|\nabla u|^{p-2} \nabla u)= \delta\, |\nabla u|^q +\mu$ in a bounded main \Om\subset\RR^n potentially with non-smooth boundary. Here either $\delta=0$ or $\delta=1$, $\mu$ is a finite signed Radon measure in $\Omega$, and $q$ is of linear or super-linear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. In particular, in the case $\delta=1$ which corresponds to a Riccati type equation, we settle the question of solvability that has been raised for some time in the literature.Comment: 18 page